# AMH2 Integrable Systems

## General Information

This page relates to the Applied Mathematics Honours course "Integrable Systems".

Lecturer for this course: Nalini Joshi and Milena Radnovic.

For general information on honours in the School of Mathematics and Statistics, refer to the relevant honours handbook.

## Course outline

Integrable Systems

An Honours Course in Applied Mathematics

*Nalini Joshi, Milena Radnović, and Yang Shi*

Weekly questions will be given below as PDF files.

Lectures delivered in the AGR will be placed here as PDF files.

The mathematical theory of integrable systems has been described as one of the most profound advances of twentieth century mathematics. They lie at the boundary of mathematics and physics and were discovered through a famous paradox that arises in a model devised to describe the thermal properties of metals (called the Fermi-Pasta-Ulam paradox).

In attempting to resolve this paradox, Kruskal and Zabusky discovered
exceptional properties in the solutions of a non-linear PDE, called the
Korteweg-de Vries equation (KdV). These properties showed that although the
solutions are waves, they interact with each other as though they were
particles, i.e., without losing their shape or speed, until then thought to
be impossible for solutions of non-linear PDEs. Kruskal invented the name
*solitons* for these solutions. Here is a picture of two solitons
interacting.

Solitons are known to arise in other non-linear PDEs and also in partial difference equations. These systems and their symmetry reductions are now called *integrable systems*. These systems occur as universal limiting models in many physical situations.

This course introduces the mathematical properties of such systems. In particular, we will study their solutions, symmetry reductions called the Painlevé equations and their discrete versions. It focuses on mathematical methods created to describe the solutions of such equations and their interrelationships. More details about the course, including course objectives and outcomes and details about the assessment and exam can be read on the Information Sheet (PDF).

- References:
- M. J. Ablowitz and H. Segur,
*Solitons and the inverse scattering transform*, SIAM, Philadelphia, USA, 1981. - M. J. Ablowitz and P.A. Clarkson,
*Solitons, nonlinear evolution equations and inverse scattering*, Cambridge University Press, Cambridge, UK, 1991. - P.G. Drazin and R.S. Johnson,
*Solitons: an introduction*, Cambridge University Press, Cambridge, UK, 1989. - M. Noumi,
*Painlevé equations through symmetry*, American Mathematical Society, Providence, R.I., USA, 2004.

- M. J. Ablowitz and H. Segur,
- Interesting Links on Solitons:
- An account of John Scott Russell's discovery of "that singular and beautiful phenomenon, which I have called the wave of translation."
- A modern attempt by mathematicians to recreate Scott Russell's wave in the Union Canal near Edinburgh.
- The Wikipedia page on Solitons (whose first and second definitions are still not correct).
- The Wikipedia page on the Korteweg-de Vries equation.

*Last modified: 5 March 2018 by Milena Radnovic*

## Online resources